Example of a mast
=================

  .. image:: apng/groundstructure.png
     :width: 7.0cm

In this example of a mast are :math:`72` nodes :math:`\left(a,b,c\right) \in \R^3`, :math:`a,b \in \{0,\frac{1}{7},\frac{2}{7}\}`, :math:`c \in \{\frac{x}{7}:x=0,1,\ldots,7\}` measured in meter and :math:`503` potential bars between every two not fixed nodes with a maximal distance of :math:`\frac{\sqrt{3}}{7}` --- long bars which are located alongside several shorter bars are ignored. The supports are in the nodes :math:`\left(a,b,0\right)`, :math:`a,b \in \{0,\frac{2}{7}\}` which are fixed in every direction. In the node :math:`\left(\frac{1}{7},\frac{1}{7},1\right)` acts a force of :math:`10^5 \, \mbox{N}` in the negative :math:`z`-direction.

:download:`The main problem is implemented as a script <mast_aufgabenstellung.py>`. The other scripts are linked by the pictures.

For a perturbation of :math:`50 \, \mbox{\%}` in every direction of the force, and scaling to obtain the same maximal quantities as in the unperturbed case, our robust optimization method produces the robust optimal solution.

Comparison of the volumes
-------------------------

+---------+----------------+----------------+
|         | optimal        | robust         |
+---------+----------------+----------------+
| volume: | 1.4000e-03     | 3.4975e-03     |
+---------+----------------+----------------+


Comparison of the volumes for different redundancies:

+------------+---------------------+---------------------+-----------------------+-----------------------+
| redundancy | :math:`\frac{1}{3}` | :math:`\frac{1}{2}` | :math:`1-\frac{1}{2}` | :math:`1-\frac{1}{3}` |
+------------+---------------------+---------------------+-----------------------+-----------------------+
| volume:    | 2.3000e-03          | 3.0000e-03          | 3.0000e-03            | 4.6000e-03            |
+------------+---------------------+---------------------+-----------------------+-----------------------+

Comparison of the volumes for different redundancies and robustness:

+------------+---------------------+-----------------------+
| redundancy | :math:`\frac{1}{2}` | :math:`1-\frac{1}{2}` |
+------------+---------------------+-----------------------+
| volume:    | 8.0805e-03          | 8.0805e-03            |
+------------+---------------------+-----------------------+

Visualizations
--------------

+------------------------------------------------------------+------------------------------------------------------------------+
| Optimal solution (:download:`script <batch_example05.py>`) | Robust solution (:download:`script <batch_example05_robust.py>`) |
+------------------------------------------------------------+------------------------------------------------------------------+
| .. image:: apng/example05.png                              | .. image:: apng/example05_robust.png                             |
|    :width: 7.0cm                                           |    :width: 7.0cm                                                 |
+------------------------------------------------------------+------------------------------------------------------------------+

Redundant solutions: (:download:`script <batch_example05_redundant.py>`)

+-----------------------------------------------+-----------------------------------------------------+
| :math:`\left(\frac{1}{3}\right)`-redundant    | :math:`\left(\frac{1}{2}\right)`-redundant          |
+-----------------------------------------------+-----------------------------------------------------+
| .. image:: apng/example05_redundant_1div3.png | .. image:: apng/example05_redundant_1div2.png       |
|    :width: 7.0cm                              |    :width: 7.0cm                                    |
+-----------------------------------------------+-----------------------------------------------------+

+-----------------------------------------------------+-----------------------------------------------------+
| :math:`\left(1-\frac{1}{2}\right)`-redundant        | :math:`\left(1-\frac{1}{3}\right)`-redundant        |
+-----------------------------------------------------+-----------------------------------------------------+
| .. image:: apng/example05_redundant_1minus1div2.png | .. image:: apng/example05_redundant_1minus1div3.png |
|    :width: 7.0cm                                    |    :width: 7.0cm                                    |
+-----------------------------------------------------+-----------------------------------------------------+

Robust and redundant solutions:

+-------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------+
| robust and :math:`\left(\frac{1}{2}\right)`-redundant: (:download:`script <batch_example05_redundant_1div2_robust.py>`) | robust and :math:`\left(1-\frac{1}{2}\right)`-redundant: (:download:`script <batch_example05_redundant_1minus1div2_robust.py>`) | 
+-------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------+
| .. image:: apng/example05_redundant_1div2_robust.png                                                                    |  .. image:: apng/example05_redundant_1minus1div2_robust.png                                                                     |
|    :width: 7.0cm                                                                                                        |     :width: 7.0cm                                                                                                               |
+-------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------+

Python scripts
--------------
* :download:`mast_aufgabenstellung.py`
* :download:`batch_example05.py`
* :download:`batch_example05_robust.py`
* :download:`batch_example05_redundant.py`
* :download:`batch_example05_redundant_1div2_robust.py`
* :download:`batch_example05_redundant_1minus1div2_robust.py`